# Difference between revisions of "Hydrodynamical models for the chaotic dripping faucet"

Original Entry by Michelle Borkin, AP225 Fall 2009

## Overview

P. Coullet, L. Mahadevan and C. Riera, Journal of Fluid Mechanics, 526, 1-17, 2005.

## Keywords

surface tension, capillary force, chaos, hydrodynamics

## Summary

This paper presents a series of mathematical models describing the chaotic dripping of a facet. First, a static droplet of water hanging at the end of a facet is discussed, then time dependence is examined with a lubrication model, chaotic behavior is investigated using a proper orthogonal decomposition (POD) resulting in a simplified model, and finally a mechanical description encompassing the main features of these detailed fluid models is presented. The mathematical models presented importantly include and explain features of the dripping faucet such as the time between drips, chaotic dripping (i.e. drip-drop behavior), and the critical point between dripping and jetting.

## Soft Matter

Before examining chaotic dripping, the authors first model a static drop and investigate the possible stable shapes. A small flow rate is assumed thus the drop will not detach form the facet unless its volume exceeds a critical volume ([itex]V < V\_{c}[/itex]) and under this condition it will also be axisymmetric. The shape of the drop is determined by the minimization of the gravitational energy and surface energy. This can be written in terms of when the forces perpendicular to the interface balance:

math

where s is the __, r is __, and z is __. The interfaces (r and z) can be written as:

math

The boundary conditions are __ where Pb is the hydrostatic pressure - this unknown is the main determinant to the different family of solutions. As shown in Figure 1(b), up to [itex]V\_{c}[/itex] the capillary forces are able to support the weight of the drop, after which the drop will be pinched-off.

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